cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A168592 G.f.: exp( Sum_{n>=1} A082758(n)*x^n/n ), where A082758(n) = sum of the squares of the trinomial coefficients in row n of triangle A027907.

Original entry on oeis.org

1, 3, 14, 80, 509, 3459, 24579, 180389, 1356743, 10402493, 81004516, 638886082, 5093081983, 40971735401, 332187974718, 2711668091448, 22267979870143, 183830653156341, 1524747465249750, 12700172705956876, 106187411693668179
Offset: 0

Views

Author

Paul D. Hanna, Dec 01 2009

Keywords

Comments

Number of lattice paths from (0,0) to (n,n) which do not go above the diagonal x=y using steps (1,k), (k,1) with k >= 0 and two kinds of (1,1). - Alois P. Heinz, Oct 07 2015
Number of pairs of noncrossing paths of length n which start and end together, each taking steps (1,0), (1,1) or (1,-1) (i.e., Motzkin-type). - Nicholas R. Beaton, Jun 17 2024

Examples

			G.f.: A(x) = 1 + 3*x + 14*x^2 + 80*x^3 + 509*x^4 + 3459*x^5 + ...
log(A(x)) = 3*x + 19*x^2/2 + 141*x^3/3 + 1107*x^4/4 + 8953*x^5/5 + ... + A082758(n)*x^n/n + ...

Links

Crossrefs

Cf. A168590, A168593, A082758, A027907, A168595, A218321, A263316.

Programs

  • Maple
    b:= proc(x, y) option remember; `if`(y<0 or y>x, 0, `if`(x=0, 1,
      add(b(x-i, y-1), i=0..x) +add(b(x-1, y-j), j=0..y)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Oct 07 2015
# second Maple program:
a:= proc(n) option remember; `if`(n<4, [1, 3, 14, 80][n+1],
      ((10*(n+1))*(16*n^3-20*n^2-n-1) *a(n-1)
      +(-944*n^4+2596*n^3-1924*n^2+236*n+30) *a(n-2)
      +(90*(n-2))*(16*n^3-52*n^2+45*n-6) *a(n-3)
      -(81*(2*n-5))*(n-2)*(n-3)*(4*n-1) *a(n-4))/
      ((n+1)*(4*n-5)*(2*n+1)*(n+2)))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Oct 07 2015
  • Mathematica
    (1/x)*InverseSeries[x*(1 - x)^2/((1 + x)^2*(1 - x + x^2)) + O[x]^30, x] // CoefficientList[#, x]& (* Jean-François Alcover, Jun 09 2018 *)
  • PARI
    {a(n)=if(n==0,1,polcoeff(exp(sum(m=1,n,sum(k=0,2*m, polcoeff((1+x+x^2)^m,k)^2)*x^m/m) +x*O(x^n)),n))}
  • PARI
    {a(n)=polcoeff(1/x*serreverse(x*(1-x)^2/((1+x)^2*(1-x+x^2)+x*O(x^n))),n)}

Formula

G.f.: A(x) = (1/x)*Series_Reversion[x*(1-x)^2/((1+x)^2*(1-x+x^2))].
G.f.: A(x) satisfies A(x^2) = M(x)*M(-x), where M(x) is the g.f. of A001006. - Alexander Burstein, Oct 03 2017
G.f.: A(x) satisfies A(x^2) = (1-x - sqrt(1-2*x-3*x^2))*(1+x - sqrt(1+2*x-3*x^2))/(4*x^4). - Paul D. Hanna, Oct 05 2017, concluded from formula of Alexander Burstein.

A168593 G.f.: exp( Sum_{n>=1} A132303(n)*x^n/n ), where A132303(n) = sum of the cubes of the trinomial coefficients in row n of triangle A027907.

Original entry on oeis.org

1, 3, 27, 349, 5484, 96408, 1824758, 36393090, 754696998, 16130052394, 353134333470, 7884110379006, 178908263232959, 4115917059924057, 95806493175049929, 2252809457441037107, 53443567449376649304
Offset: 0

Views

Author

Paul D. Hanna, Dec 01 2009

Keywords

Comments

Self-convolution cube-root yields the integer sequence A251686.

Examples

			G.f.: A(x) = 1 + 3*x + 27*x^2 + 349*x^3 + 5484*x^4 + 96408*x^5 +...
log(A(x)) = 3*x + 45*x^2/2 + 831*x^3/3 + 17181*x^4/4 + 375903*x^5/5 +...+ A132303(n)*x^n/n +...

Crossrefs

Cf. A168590, A168592, A132303, A027907, A251686.

Programs

  • PARI
    {A027907(n,k) = polcoeff((1+x+x^2)^n, k)}
{A132303(n) = sum(k=0, 2*n, A027907(n,k)^3)}
{a(n) = local(A); A = exp(sum(m=1, n+1, A132303(m)*x^m/m) +x*O(x^n)); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

A168591 a(n) = sum of the n-th power of the trinomial coefficients in row n of triangle A027907.

Original entry on oeis.org

1, 3, 19, 831, 281907, 764206503, 16955359883149, 3120135986465328015, 4819828380706011142058787, 63004580363300194268932114354503, 7017256966823394610075464951176481843849
Offset: 0

Views

Author

Paul D. Hanna, Dec 01 2009

Keywords

Crossrefs

Cf. A168590, A027907.

Programs

  • PARI
    a(n)=sum(k=0,2*n,polcoeff((1+x+x^2)^n,k)^n)

Formula

Ignoring initial term, equals the logarithmic derivative of A168590.

A198203 G.f.: exp( Sum_{n>=0} [ Sum_{k=0..2*n} A027907(n,k)^n * x^k ]* x^n/n ), where A027907 is the triangle of trinomial coefficients.

Original entry on oeis.org

1, 1, 2, 5, 19, 160, 3418, 179705, 19488053, 4590422901, 2738580784946, 3583015072969210, 9255051219746866753, 56916338252385095986978, 871826913772059843867743765, 26753845554560439025697319191184, 1695956186616651065722319776300825712
Offset: 0

Views

Author

Paul D. Hanna, Oct 22 2011

Keywords

Comments

Trinomial coefficients satisfy: Sum_{k=0..2*n} A027907(n,k)*x^k = (1+x+x^2)^n.

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 19*x^4 + 160*x^5 + 3418*x^6 +...
The logarithm begins:
log(A(x)) = x + 3*x^2/2 + 10*x^3/3 + 55*x^4/4 + 691*x^5/5 + 19440*x^6/6 + 1232750*x^7/7 + 154436735*x^8/8 + 41136723397*x^9/9 +...
which equals the sum of the series:
log(A(x)) = (1 + x + x^2)*x
+ (1 + 2^2*x + 3^2*x^2 + 2^2*x^3 + x^4)*x^2/2
+ (1 + 3^3*x + 6^3*x^2 + 7^3*x^3 + 6^3*x^4 + 3^3*x^5 + x^6)*x^3/3
+ (1 + 4^4*x + 10^4*x^2 + 16^4*x^3 + 19^4*x^4 + 16^4*x^5 + 10^4*x^6 + 4^4*x^7 + x^8)*x^4/4
+ (1 + 5^5*x + 15^5*x^2 + 30^5*x^3 + 45^5*x^4 + 51^5*x^5 + 45^5*x^6 + 30^5*x^7 + 15^5*x^8 + 5^5*x^9 + x^10)*x^5/5 +...

Crossrefs

Cf. A186236, A168590, A027907.

Programs

  • PARI
    {A027907(n, k)=polcoeff((1+x+x^2)^n, k)}
{a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, 2*m, A027907(m, k)^m *x^k) *x^m/m)+x*O(x^n)), n)}
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.